This novel book introduces cellular automata from a rigorous nonlinear dynamics perspective. It supplies the missing link between nonlinear differential and difference equations to discrete symbolic analysis. A surprisingly useful interpretations of cellular automata in terms of neural networks is also given. The book provides a scientifically sound and original analysis, and classifications of the empirical results presented in Wolfram s monumental New Kind of Science.

Volume 2: From Bernoulli Shift to 1/f Spectrum; Fractals Everywhere; From Time-Reversible Attractors to the Arrow of Time; Mathematical Foundation of Bernoulli -Shift Maps; The Arrow of Time; Concluding Remarks.

Perspective of Wolfram’s New Kind of Science Table 2. (Continued ) Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 2. (Continued ) 417 418 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 2. (Continued ) Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 2. (Continued ) 419 420 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 2. (Continued ) Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 2. (Continued ) 421 422

(Continued ) 427 428 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 2. (Continued ) Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 2. (Continued ) 429 430 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 2. (Continued ) Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 2. (Continued ) 431 432 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 2. (Continued ) Chapter 4: From Bernoulli Shift to

2 that all eight noninvertible and nonbilateral period-2 rules in Table 8 possess an additional form of symmetry; namely, the forward time-1 map ρ1 [N ] is related to the backward time-1 map ρ†1 [N ] by a 180◦ rotation about the center. In other words, the left and right vignette frames of each rule in Table 8 are related by a 180◦ rotation about the origin. We will henceforth call this rather rare property a self π-rotation symmetry. 4.3. Period-3 rules A comprehensive examination of Table 2

four rules belong to the same global equivalence class ε222 , as depicted in Table 9, it follows that it suﬃces to conduct an in-depth analysis of only one of these four rules. Since we have already been exposed to 62 in Fig. 11, let us continue to use this rule for illustrations. Figure 16 shows the dynamic pattern D N [x(0)] of 62 , 118 , 131 , and 145 for diﬀerent choices of initial states which give rise to qualitatively similar evolution patterns. Observe that each pattern converges to a

“shift left by 2 pixels”. 464 A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 10. 84 Invertible Bernoulli στ -shift rules with one or two Bernoulli attractors. Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 11. 20 Noninvertible Bernoulli στ -shift rules with two Bernoulli attractors. Table 12. 8 Noninvertible Bernoulli στ -shift rules with three Bernoulli attractors. column due to space limitation. Observe that there are four distinct Bernoulli στ -shift